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Nonlinear convex and concave relaxations for the solutions of parametric ODEs

Authors

  • Joseph K. Scott,

    1. Process Systems Engineering Laboratory, Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, 02139, USA
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  • Benoit Chachuat,

    1. Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College, London, SW7 2AZ, UK
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  • Paul I. Barton

    Corresponding author
    • Process Systems Engineering Laboratory, Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, 02139, USA
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Correspondence to: Paul I. Barton, Process Systems Engineering Laboratory, Department of Chemical Engineering, Massachusetts Institute of Technology, 66-464, 77 Massachusetts Ave., Cambridge, MA 02139, USA.

E-mail: pib@mit.edu

SUMMARY

Convex and concave relaxations for the parametric solutions of ordinary differential equations (ODEs) are central to deterministic global optimization methods for nonconvex dynamic optimization and open-loop optimal control problems with control parametrization. Given a general system of ODEs with parameter dependence in the initial conditions and right-hand sides, this work derives sufficient conditions under which an auxiliary system of ODEs describes convex and concave relaxations of the parametric solutions, pointwise in the independent variable. Convergence results for these relaxations are also established. A fully automatable procedure for constructing an appropriate auxiliary system has been developed previously by the authors. Thus, the developments here lead to an efficient, automatic method for computing convex and concave relaxations for the parametric solutions of a very general class of nonlinear ODEs. The proposed method is presented in detail for a simple example problem. Copyright © 2012 John Wiley & Sons, Ltd.

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